FREE UNDAMPED AND DAMPED VIBRATIONS Lab Report.

ABSTRACT

A mechanical system is said to be vibrating when its component part are undergoing periodic

oscillations about a central statical equilibrium position. Any system can be caused to vibrate by

externally applying forces due to its inherent mass and elasticity.

OBJECTIVES

To investigate the response and behaviour of a pendulum system undergoing free vibrations

with and without viscous damping.

To determine values of damping coefficient ‘C’ and damping ratio for a set of damper

setting.

To verify the suitability of the mathematical method used in determining the damping

values.

DESCRIPTION OF EQUIPMENTS

1. The oscilloscope: is basically a graph-displaying device - it draws a graph of an electrical signal. It

is used in observing constantly varying signal voltages, usually as a two-dimensional graph of one

or more electrical potential differences using the vertical 'Y' axis, plotted as a function of time.

Fig 1: An Oscilloscope.

2. Potentiometer: is a three-terminal resistor with a sliding contact that forms an

adjustable voltage divider. It is a simple electro-mechanical transducer. It converts rotary or

linear motion from the operator into a change of resistance. It has terminals which can be

connected to a signal amplifying or display unit as the case may be (sound.westhost.com).

Fig 2: The Potentiometer.

3. Variable damping unit: this is used to set varying damping values been used for the experiment.

Fig 3: Damper setting unit.

4. Digital weighing scale: a measurement device used to measure the weight or mass of an object

or substances. Most digital scales make their measurements based on an internal strain gauge, a

thin foil piece that conducts electricity and is sensitive to deformation is attached with some

adhesive to a flexible surface. When weight is applied to the digital scale, various mechanisms

within the digital scale ensure the weight is evenly distributed on the strain gauge. The weight

bends the flexible surface, deforming the foil piece, which alters the flow of electrical current

(www.wisegeek.com).

Fig 4: The weighing scale.

5. Helical Spring: an elastic body, whose function is to distort when loaded and to recover

its original shape when the load is removed. It is made up of a wire coiled in the form of a

helix and are primarily intended for compressive or tensile loads (engg-learning.blogspot.com).

Fig 5: Attached helical spring.

6. Signal Amplifier: is an electronic device that increases the power of a signal. It does this by taking

energy from a power supply and controlling the output to match the input signal shape but with

a larger amplitude. In this sense, an amplifier modulates the output of the power supply

(en.wikipedia.org).

Fig 6: The signal amplifier.

The mass is mounted at one end of the arm of the horizontally mounted pendulum with the other

end connected by a helical spring to a fixed support. A rotary potentiometer is mounted on the

hinged end of the pendulum. The potentiometer receives the displacement signal as the pendulum

swings and this signal is transmitted through a signal amplifier connected to the oscilloscope to be

displayed.

Fig 7: Set up of the equipment for the laboratory exercise.

THEORETICAL ANALYSIS

If a mechanical system is displaced from its equilibrium position and then released, the restoring

force will bring about return towards the equilibrium position. This is referred to as “Free Vibration”.

This type of vibration arises from an initial impact energy that is continually changing from potential

to kinetic form. In a free vibration, the system is said to vibrate at its natural frequency. However,

due to various causes there will be some dissipation of mechanical energy during each cycle of

vibration and this effect is called “Damping.” (Ryder and Bennett, 1990).

Theoretically, an un-damped free vibration system continues vibrating once it is started. This

experiment examines the effect of damping and the level of damping on the behaviour of a

pendulum.

Vibration can be classified in several ways, the important ones includes:

Free and Forced Vibration: in this type of vibration, no external force acts on the system as

the system is left to vibrate on its own after an initial disturbance. E.g.: the oscillation of a

simple pendulum. Similarly, when the system is subjected to an external force (often, a

repeating type of force. E.g.: the oscillation of machines like diesel engines.

Undamped and Damped Vibration: if no energy is lost or dissipated in friction or resistance

during oscillation, the vibration is known as undamped vibration. Should any energy be lost

in its way, it is called damped vibration. (S.S Rao. 2011).

Other ways of classifying vibration are; linear and non-linear vibration, deterministic and random

vibration.

The simplest possible vibratory system consist of a mass attached by means of a spring to an

immovable support as shown below. The mass is constrained to translational motion in the direction

of the axis so that its change of position from an initial reference is described fully by the value of

a single quantity . This is called a ‘single degree of freedom’ (R.E Blake, 2002).

Fig 8: Undamped single degree of freedom system.

The differential equation of motion of mass, m for the undamped system is:

The angular natural frequency is given by: √

rad/sec

Here: k – spring stiffness m – mass.

In order to simplify the mathematics involved, the damping is modelled as a viscous damping

depending on the magnitude of damping. A damped system can be under-damped, critically damped

or over-damped.

Fig 9: Damped system model.

For a damped system, the corresponding equation of motion of mass is given by:

Under-damped System: this occurs when the damping of the system is less than critical, ζ<1; a

simple analogy is the underdamped door closer would close quickly, but would hit the door frame

with significant velocity, or would oscillate in the case of a swinging door. The solution of equation

above is:

⁄ ( )

⁄ ( )

The damped natural frequency for the vibration is:

√

Fig 10: Typical response to a step disturbance of an under-damped system.

Critically Damped System: If the damping is increased, the oscillations die away quicker and

eventually a critical point is reached where the mass just returns to the rest position with no

overshoot or oscillation. This occurs when ζ = 1. A critically damped system converges to zero as fast

as possible without oscillating. An example of critical damping is the door closer seen on many

hinged doors in public buildings.

Fig 11: Typical response to Critically-damped system (D.V Hutton 1981).

Overdamped System: This occurs when ζ > 1. The result is an exponential decay with no oscillations

but it will take longer to reach the rest position than with critical damping. An over-damped door-

closer will take longer to close than a critically damped door would.

Fig 12: Typical response to an Over-damped system (D.V Hutton 1981).

Logarithmic Decrement: the displacement of an underdamped system is a sinusoidal oscillation with

decaying amplitude. A quite useful property of an underdamped system can be obtained by

comparing the amplitudes of any two successive cycles of the displacement (D.V. Hutton, 1981).

( )

Here:

is the period of the motion

Substitution of T and in the equation above:

√

The term

√

is called the logarithmic decrement of the response.

Mathematical model

To determine the values of spring stiffness, K and pendulum arm mass, Ma using the following

equations:

√

√

Where:

Large Mass, ML Small Mass, MS

FL – Natural frequency with ML attached to arm

FS – Natural frequency with MS attached to arm

K – Spring stiffness Ma – Pendulum arm mass.

(√

) --------------- (1)

(√

) --------------- (2)

From equation (1) Square both side

(

)

( ) (

)

( ) ( ) ( )

From equation (2) Square both sides

(

)

( ) (

)

( ) ( ) ( )

( )

( ) ( ) ( ) ( )

( )

( )

(

)

Therefore;

Part B: To find the value of damper coefficient .

(

)

√

But (

)

√

√

( )√

( ) ( )

( ) ( )

( ) ( )

( ) ( ( ) )

( )

( ( ) )

√( )

( )

√

√( )

EXPERIMENTAL METHOD AND CALCULATIONS

Part A.

i. At commencing the experiment, the value (weight) of the small mass and large masses were

ascertained.

ii. One of the available masses was attached to the end of the tube which forms the pendulum

arm.

iii. The oscilloscope is set to ‘single arm mode.’ After a suitable voltage sensitivity setting was

achieved, the pendulum is made to swing freely against the spring.

iv. The vibration was noted and the frequency reading was recorded from the slope.

v. The procedure was repeated twice and the average value for the frequency was rightly

noted. Similarly, the values of the frequency was determined for using the second mass and

the also when attached with no mass.

Part B

i. With initial apparatus set up in ‘part A’ previously, the large mass was attached and also the

damper unit to pendulum arm.

ii. The damper unit was set to 1 on the damper scale and the scope was adjusted to read zero

voltage by moving the vertical cursor and aligning the horizontal cursor on the input signal

trace.

iii. The pendulum was made to swing freely against the spring with the resulting trace on the

scope fixed using the store facility.

iv. The voltage reading which represented the peak value of the waveforms were noted. This

step is repeated for damper scale setting of 3 and 5.

v. The damper was set to a higher scale and the corresponding system response was observed.

Experimental Values 1 2 3 Average

Natural Frequency, fL with mL attached to arm (Hz) 3.73 3.68 3.73 3.71

Natural Frequency, fS with mS attached to arm (Hz) 4.31 4.39 4.39 4.36

Natural Frequency, f0 without mass 5.68 5.68 5.68 5.68

Calculations

( ) ( )

( ) ( )

Sub for = 0.0825 in equation (4)

( ) ( )

K = 39.47 × 4.362 × (0.0825 + 0.0647)

K = 39.47 × 19.01 × 0.1472

K = 110.45 Kg/m2

F0 – natural frequency without mass = 5.68Hz

To check for the value of the natural frequency F0 of the arm:

(√

)

(√

)

F0 = 0.159 × 36.59

F0 = 5.82Hz

Successive amplitude ratio for damper setting 1

Average value of amplitude ratio for setting 1

Successive amplitude ratio for damper setting

Average value of amplitude ratio for setting 3

Successive amplitude ratio for damper setting 5

Average value of amplitude ratio for setting 5

To determine the values of the damping ratio, ε for each damper setting.

√( )

( )

Damping setting = 1.

√( )

( ) √

Damper setting = 3.

√( )

( ) √

Damper setting = 5.

√( )

( ) √

To determine the values of the damping coefficient, c for each damper setting

√( )

M = MA + ML

M = 0.0825 + 0.1208 = 0.2033Kg

Damper setting = 1.

√ √

Damper setting = 3.

√ √

Damper setting = 5.

√ √

RESULTS AND ANALYSIS.

Damper Setting

X1

X2

X3

X4

X1

X2

X3

X4

20.00 16.0 15.3 13.3

1 37.3 28.0 26.0 24.0 93.3 63.3 59.9 54.6

26.0 19.3 18.6 17.3

36.0 24.0 22.0 19.3

3 23.3 16.6 15.3 14.0 83.3 57.2 53.3 47.9

24.0 16.6 16.0 14.6

22.0 13.3 12.6 11.3

5 20.6 14.0 12.6 11.3 67.9 45.9 40.5 35.9

25.3 18.6 15.3 13.3

Experimental

Values

Calculated Values

F0 = 5.68 F0 = 5.82

MA = 0.0825Kg/m2

K = 110.45Kg/m

Damper Setting Damping coefficient, c Damping ratio,

1 0.218 0.023

3 0.294 0.031

5 0.332 0.035

Analysis: A vibratory system is a dynamic one which for which the variables such as the

excitations (input) and responses (output) are time dependent. The response of a vibrating system

generally depends on the initial conditions as well as any form of external excitations (S.S Rao,

2011). The vibrations which occur in a mechanical equipment most often results from forces which

arise from the functional operation of the equipment (D.V. Hutton, 1981). Therefore, analysing a

vibrating system will involve setting up a mathematical model, deriving and solving equations

pertaining to the model, interpreting the results and assumptions and reanalyse or redesign if need

be.

DISCUSSION.

The experimental and calculated results does not differ much as there is a marginal error of about

2.4% (that is:(

) ). This might be as a result of some system imbalance

and/or hysteresis. Similarly, the discrepancy might be a result of some error in calculations.

As the experiment was been carried out with different damper settings, the frequency of oscillation

of the pendulum reduces as the damper setting increases. When the damper was set to a high

setting, the pendulum simply remained stationary because the there was no room for any form of

displacement of the pendulum except there is additional force exerted to overcome the damping

force in place.

The mathematical model used for the system is a valid one for determining the damping values as it

was not too complex. Starting with an initial elementary model of differential equation of motion

and then developed gradually and refined to accommodate the other input components and details

to closely calculate and observe the system behaviour.

Advantages of damping

Dampers dissipate energy within a system by converting it to heat. If designed properly, damping

forces can be completely out of phase with structural stress. Thus, the right damper can reduce

stress and deflection simultaneously. Without damping, there would be no suspension in cars. With

no suspensions, it would be twice as dangerous to drive in a car as it is today. The handling of the

wheels would be extremely difficult, and breaking will be much uneasy.

Damping is employed across different areas aside the automotive industry which includes but not

limited to Aerospace and Defense, Heavy Industry steel Mills, Aluminium Mills, Shipbuilders

Offshore Oil Drilling, Civil Engineering buildings, Bridges and Stadiums Towers.

Similarly, Damping can cause additional friction losses and heat build-up in certain machineries

which a bit of disadvantage.

CONCLUSION

Set objectives for this laboratory exercise was achieved with a good knowledge about the subject

matter of pendulum behaviour and response when subjected to free oscillatory motions.

The theory of free vibrational motion with and without viscous damping was studied and

appropriate mathematical model was used to calculate the value of the spring stiffness, K the

natural frequency, when the large mass was attached, also the value of the natural frequency,

with no mass attached. The damping coefficient was also calculated for different damping setting.

Damping is very useful and it should be incorporated in the design of systems or mechanism

subjected to vibrations and shock as it helps to minimize fatigue and failure. The right damper will

reduce stress and deflection.

REFERENCES:

David H. Hutton, Applied Mechanical Vibrations (1981), McGraw-Hill Series. London.

G.H. Ryder and M.D. Bennett, Mechanics of Machines (2nd Ed), Macmillan: Hong-Kong.

Singiresu S. Rao, Mechanical Vibrations (5th Ed.) 2011, Pearson: Singapore.

Douglas P. Taylor, The Application of Energy Dissipating Damping devices to an Engineered

Structure or Mechanism [Online].http://www.shockandvibration.comn

Accessed: 20th March 2014.

JDJ: Vibration Types [Online]

http://www.mcasco.com/Answers/qa_vtype.htm

accessed: 20th March 2014.